Explicit expression for coefficients of arbitrary integer powers of power series?

Question

For $n=2,4,6,\dots$, define $$ f_n(x)=\left(\sum_{k=0}^\infty a_k x^k\right)^n=\sum_{k=0}^\infty c_k x^k. $$ Certainly, for $n=2$ we have the Cauchy product $$ f_2(x)=\sum_{k=0}^\infty \sum_{\ell=0}^ka_{k-\ell}a_\ell x^k. $$ For higher orders of $n$ I am aware that we also have the Faà di Bruno's formula. My main interest is in an explicit expression for the coefficients $c_k$ for arbitrary $n$. A recurrence relation would be fine as well. My hope is that this has already been worked out and I will not have to drudge through Faà di Bruno to work it out from scratch.